Connections Between Mirror Descent, Thompson Sampling and the Information Ratio

The information-theoretic analysis by Russo and Van Roy (2014) in combination with minimax duality has proved a powerful tool for the analysis of online learning algorithms in full and partial information settings. In most applications there is a tantalising similarity to the classical analysis based on mirror descent. We make a formal connection, showing that the information-theoretic bounds in most applications can be derived from existing techniques for online convex optimisation. Besides this, for $k$-armed adversarial bandits we provide an efficient algorithm with regret that matches the best information-theoretic upper bound and improve best known regret guarantees for online linear optimisation on $\ell_p$-balls and bandits with graph feedback

First-Order Regret Analysis of Thompson Sampling

We address online combinatorial optimization when the player has a prior over the adversary's sequence of losses. In this framework, Russo and Van Roy proposed an information-theoretic analysis of Thompson Sampling based on the {\em information ratio}, resulting in optimal worst-case regret bounds. In this paper we introduce three novel ideas to this line of work. First we propose a new quantity, the scale-sensitive information ratio, which allows us to obtain more refined first-order regret bounds (i.e., bounds of the form $\sqrt{L^*}$ where $L^*$ is the loss of the best combinatorial action). Second we replace the entropy over combinatorial actions by a coordinate entropy, which allows us to obtain the first optimal worst-case bound for Thompson Sampling in the combinatorial setting. Finally, we introduce a novel link between Bayesian agents and frequentist confidence intervals. Combining these ideas we show that the classical multi-armed bandit first-order regret bound $\tilde{O}(\sqrt{d L^*})$ still holds true in the more challenging and more general semi-bandit scenario. This latter result improves the previous state of the art bound $\tilde{O}(\sqrt{(d+m^3)L^*})$ by Lykouris, Sridharan and Tardos.Comment: 27 page

Personalized Federated Learning with Hidden Information on Personalized Prior

Federated learning (FL for simplification) is a distributed machine learning technique that utilizes global servers and collaborative clients to achieve privacy-preserving global model training without direct data sharing. However, heterogeneous data problem, as one of FL's main problems, makes it difficult for the global model to perform effectively on each client's local data. Thus, personalized federated learning (PFL for simplification) aims to improve the performance of the model on local data as much as possible. Bayesian learning, where the parameters of the model are seen as random variables with a prior assumption, is a feasible solution to the heterogeneous data problem due to the tendency that the more local data the model use, the more it focuses on the local data, otherwise focuses on the prior. When Bayesian learning is applied to PFL, the global model provides global knowledge as a prior to the local training process. In this paper, we employ Bayesian learning to model PFL by assuming a prior in the scaled exponential family, and therefore propose pFedBreD, a framework to solve the problem we model using Bregman divergence regularization. Empirically, our experiments show that, under the prior assumption of the spherical Gaussian and the first order strategy of mean selection, our proposal significantly outcompetes other PFL algorithms on multiple public benchmarks.Comment: 19 pages, 6 figures, 3 table

Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems

Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the Thirties, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In this survey, we focus on two extreme cases in which the analysis of regret is particularly simple and elegant: i.i.d. payoffs and adversarial payoffs. Besides the basic setting of finitely many actions, we also analyze some of the most important variants and extensions, such as the contextual bandit model.Comment: To appear in Foundations and Trends in Machine Learnin